Through study of the Hermitian least squates solutions of A₂XA*₂ = C₂, under the limits of min x=x* ‖ A₁XB₁-C₁ ‖.By applying the rank and index of inertia of matrix function,we derive the equivalent condition about Hermitian least squares solution of A₂XA*₂ = C₂, and the solution of matrix inequality A*₂A₂XA*₂A₂>(<,≥,≤)A*₂C₂A₂ ,In particular case,we study the existence problem about the (half) positive(negative) definite's Hermitian least squares solution of A₁XB₁ = C₁.